Integrand size = 20, antiderivative size = 55 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{3} a^2 c x^3+\frac {1}{5} a (2 b c+a d) x^5+\frac {1}{7} b (b c+2 a d) x^7+\frac {1}{9} b^2 d x^9 \]
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Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459} \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{3} a^2 c x^3+\frac {1}{7} b x^7 (2 a d+b c)+\frac {1}{5} a x^5 (a d+2 b c)+\frac {1}{9} b^2 d x^9 \]
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Rule 459
Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 c x^2+a (2 b c+a d) x^4+b (b c+2 a d) x^6+b^2 d x^8\right ) \, dx \\ & = \frac {1}{3} a^2 c x^3+\frac {1}{5} a (2 b c+a d) x^5+\frac {1}{7} b (b c+2 a d) x^7+\frac {1}{9} b^2 d x^9 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{3} a^2 c x^3+\frac {1}{5} a (2 b c+a d) x^5+\frac {1}{7} b (b c+2 a d) x^7+\frac {1}{9} b^2 d x^9 \]
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Time = 2.62 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {b^{2} d \,x^{9}}{9}+\frac {\left (2 a b d +b^{2} c \right ) x^{7}}{7}+\frac {\left (a^{2} d +2 a b c \right ) x^{5}}{5}+\frac {a^{2} c \,x^{3}}{3}\) | \(52\) |
norman | \(\frac {b^{2} d \,x^{9}}{9}+\left (\frac {2}{7} a b d +\frac {1}{7} b^{2} c \right ) x^{7}+\left (\frac {1}{5} a^{2} d +\frac {2}{5} a b c \right ) x^{5}+\frac {a^{2} c \,x^{3}}{3}\) | \(52\) |
gosper | \(\frac {1}{9} b^{2} d \,x^{9}+\frac {2}{7} x^{7} a b d +\frac {1}{7} x^{7} b^{2} c +\frac {1}{5} x^{5} a^{2} d +\frac {2}{5} x^{5} a b c +\frac {1}{3} a^{2} c \,x^{3}\) | \(54\) |
risch | \(\frac {1}{9} b^{2} d \,x^{9}+\frac {2}{7} x^{7} a b d +\frac {1}{7} x^{7} b^{2} c +\frac {1}{5} x^{5} a^{2} d +\frac {2}{5} x^{5} a b c +\frac {1}{3} a^{2} c \,x^{3}\) | \(54\) |
parallelrisch | \(\frac {1}{9} b^{2} d \,x^{9}+\frac {2}{7} x^{7} a b d +\frac {1}{7} x^{7} b^{2} c +\frac {1}{5} x^{5} a^{2} d +\frac {2}{5} x^{5} a b c +\frac {1}{3} a^{2} c \,x^{3}\) | \(54\) |
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Time = 0.29 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{9} \, b^{2} d x^{9} + \frac {1}{7} \, {\left (b^{2} c + 2 \, a b d\right )} x^{7} + \frac {1}{3} \, a^{2} c x^{3} + \frac {1}{5} \, {\left (2 \, a b c + a^{2} d\right )} x^{5} \]
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Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {a^{2} c x^{3}}{3} + \frac {b^{2} d x^{9}}{9} + x^{7} \cdot \left (\frac {2 a b d}{7} + \frac {b^{2} c}{7}\right ) + x^{5} \left (\frac {a^{2} d}{5} + \frac {2 a b c}{5}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{9} \, b^{2} d x^{9} + \frac {1}{7} \, {\left (b^{2} c + 2 \, a b d\right )} x^{7} + \frac {1}{3} \, a^{2} c x^{3} + \frac {1}{5} \, {\left (2 \, a b c + a^{2} d\right )} x^{5} \]
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Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=\frac {1}{9} \, b^{2} d x^{9} + \frac {1}{7} \, b^{2} c x^{7} + \frac {2}{7} \, a b d x^{7} + \frac {2}{5} \, a b c x^{5} + \frac {1}{5} \, a^{2} d x^{5} + \frac {1}{3} \, a^{2} c x^{3} \]
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Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int x^2 \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx=x^5\,\left (\frac {d\,a^2}{5}+\frac {2\,b\,c\,a}{5}\right )+x^7\,\left (\frac {c\,b^2}{7}+\frac {2\,a\,d\,b}{7}\right )+\frac {a^2\,c\,x^3}{3}+\frac {b^2\,d\,x^9}{9} \]
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